An Analog of the Cauchy-Schwarz Inequality for Hadamard Products and Unitarily Invariant Norms

We show that for any unitarily invariant norm k k on M n (the space of n-by-n complex matrices) where denotes the Hadamard (entrywise) product. These results are a consequence of an inequality for absolute norms on C n kx yk 2 kx xk ky yk for all x; y 2 C n : (2) We also characterize the norms on C n that satisfy (2), characterize the unitary similarity invariant norms on M n that satisfy (1), and obtain related results on norms on C n and unitary similarity invariant norms on M n that are of independent interest.